Originally Posted by **rgep**

I think you're mixing up the relation and the classes.

Any relation can be expressed as a set of ordered pairs, where (a,b) is in R if and only if the relation aRb is true. So equality is expressed as the set (1,1),(2,2),(3,3) ... Similarly the relation < can be expressed as the set (1,2),(1,3),(1,4),...(2,3),(2,4),...(3,4),... .

If a relation is an equivalence (RST: reflexive, symmetric, transitive) then it can also be defined by the classes of things which are equivalent. For equality this is the collection of singletons {1}, {2}, {3}, ...

Congruence modulo 2 is also known as parity: a==b iff their difference a-b is even. The relation is expressed as (1,1),(1,3),(1,5),...,(3,3,),(3,5),...,(2,2),(2,4) ,(2,6),...,(4,4),(4,6),... and the two classes are {1,3,5,...} and {2,4,6,...}.